Let Q be a differential operator of order ≤1\documentclass[12pt]{minimal}
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\begin{document}$$\le 1$$\end{document} on a complex metric vector bundle E→M\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {E}\rightarrow \mathscr {M}$$\end{document} with metric connection ∇\documentclass[12pt]{minimal}
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\begin{document}$$\nabla $$\end{document} over a possibly noncompact Riemannian manifold M\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {M}$$\end{document}. Under very mild regularity assumptions on Q that guarantee that ∇†∇/2+Q\documentclass[12pt]{minimal}
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\begin{document}$$\nabla ^{\dagger }\nabla /2+Q$$\end{document} canonically induces a holomorphic semigroup e-zHQ∇\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {e}^{-zH^{\nabla }_{Q}}$$\end{document} in ΓL2(M,E)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _{L^2}(\mathscr {M},\mathscr {E})$$\end{document} (where z runs through a complex sector which contains [0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$[0,\infty )$$\end{document}), we prove an explicit Feynman–Kac type formula for e-tHQ∇\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {e}^{-tH^{\nabla }_{Q}}$$\end{document}, t>0\documentclass[12pt]{minimal}
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\begin{document}$$t>0$$\end{document}, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact M\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {M}$$\end{document}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Tr}\left( \widetilde{V}\int ^t_0\mathrm {e}^{-sH^{\nabla }_{V}}P\mathrm {e}^{-(t-s)H^{\nabla }_{V}}\mathrm {d}s\right) , \end{aligned}$$\end{document}where V,V~\documentclass[12pt]{minimal}
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\begin{document}$$V,\widetilde{V}$$\end{document} are of zeroth order and P is of order ≤1\documentclass[12pt]{minimal}
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\begin{document}$$\le 1$$\end{document}. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.
